Today, we learn not to turn our back on
information. The University of Houston's College of
Engineering presents this series about the machines
that make our civilization run, and the people
whose ingenuity created them.
I've been running into the
Monty Hall Problem lately. I suspect that
many of you know about it. It came to my attention
the other day when I ran into a colleague from the
math department. She told me about it and left me
scoffing in disbelief.
I should've known about the problem, since it goes
back to the old TV show, Let's Make a Deal.
Host Monty Hall would offer a contestant three
doors. One had a prize behind it. If the contestant
guessed the correct door, he would win the prize.
But, before the door that he chose was opened,
Monty Hall (who knew where the prize was) would
say, "Of the two remaining doors, I'll open this
one, which has no prize behind it." Then Hall would
add, "Now, would you like to change your guess?"
The contestant could either decide that the first
guess was correct or switch to the other unopened
door.
"The contestant should switch," said my
mathematician friend. "Why?" I asked. "Because the
probability of getting the prize will rise from one
chance in three to two out of three."
A couple of really strange issues lurk here. First
of all, there is the logic of probability. Second,
we're trained from childhood that it's good to be
decisive -- to be resolute. It strikes us as
wishy-washy to change our minds.
Let's put the matter of decisiveness aside and look
at probability. Our instinct says there's now a
fifty-fifty chance of finding the prize behind
either remaining door. Why switch? It seems
preposterous that the probability of the prize's
being behind the other unopened door should double
after Hall opens that third door.
But the likelihood of the prize's being where you
first thought it was is still one in three. That
was true before Hall opened that other door, and it
remains so. The chance that the prize was behind
one of the other two doors was two-thirds, and it
also remains so. But that two-thirds chance has now
been given to the single remaining door. The
contestant absolutely does need to switch choices.
This riddle has a point that lies just beyond our
line of sight. It warns us that information is good
in ways we don't immediately see. When Hall opens
that door, he gives us information that serves us
in unexpected ways.
That's how probability theory serves us. There's
almost always more information in the facts before
us than we realize. We tend to be unaware of how
much information is being condensed, all the time,
within our telephone and computer systems. It's
done by the use of incredibly sophisticated
statistical theory. Every shred of information is
being wrung out and used to the hilt.
On the one hand, we cry out to brush the Monty Hall
problem off as "lying with statistics." But our
electronic systems would break down without that
sort of thinking. The Monty Hall Problem is a
powerful reminder that information really does gain
the prize behind the door -- when we know how to
use it.
I'm John Lienhard, at the University of Houston,
where we're interested in the way inventive minds
work.
(Theme music)
My thanks to both Barbara Keyfitz and Martin
Golubitsky, UH Math Department, for their counsel.
The reader who is unconvinced might want to look at
one of the excellent web sites that turn up when
one searches for the "Monty Hall Problem." Several
of these allow one to test the result empirically
by playing the game several times in rapid
succession. One might want to look at the excellent
mathematical explanation of the problem by D. P.
Bertsekas and John N. Tsitsiklis, Introduction
to Probability. Belmont, MA: Athena
Scientific, 2002, Section 1.3.
And finally, Aiden Keefe writes to point out that
all this applies only if Monty Hall offers the
choice every time. If he offers the choice only
when the contestant had already chosen incorrectly,
he is then doing the contestant no favor.
The Engines of Our Ingenuity is
Copyright © 1988-2000 by John H.
Lienhard.
